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Modern challenge to Algorithm Design Data = Massive; Computers = Tiny Data = Massive; Computers = Tiny How can tiny computers analyze massive data? How can tiny computers analyze massive data? Only option: Design sublinear time algorithms. Only option: Design sublinear time algorithms. Algorithms that take less time to analyze data, than it takes to read/write all the data. Algorithms that take less time to analyze data, than it takes to read/write all the data. Can such algorithms exist? Can such algorithms exist? December 2, 2009 IPAM: Invariance in Property Testing 2

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Yes! Polling … Is the majority of the population Red/Blue Is the majority of the population Red/Blue Can find out by random sampling. Can find out by random sampling. Sample size / margin of error Sample size / margin of error Independent of size of population Independent of size of population Other similar examples: (can estimate other moments …) Other similar examples: (can estimate other moments …) December 2, 2009 IPAM: Invariance in Property Testing 3

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My concerns … Why is the understanding of Algebraic Property Testing so far behind? Why is the understanding of Algebraic Property Testing so far behind? Why can’t we get “rich” class of properties that are all testable? Why can’t we get “rich” class of properties that are all testable? Why are proofs so specific to property being tested. Why are proofs so specific to property being tested. What made Graph Property Testing so well- understood? What made Graph Property Testing so well- understood? What is “novel” about Property Testing, when compared to “polling”? What is “novel” about Property Testing, when compared to “polling”? December 2, 2009 IPAM: Invariance in Property Testing 17

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Invariances are the key? “Polling” works well when (because) invariant group of property is the full symmetric group. “Polling” works well when (because) invariant group of property is the full symmetric group. Modern property tests work with much smaller group of invariances. Modern property tests work with much smaller group of invariances. Graph property ~ Invariant under vertex renaming. Graph property ~ Invariant under vertex renaming. Algebraic Properties & Invariances? Algebraic Properties & Invariances? December 2, 2009 IPAM: Invariance in Property Testing 20

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Abstracting Algebraic Properties [Kaufman & S.] [Kaufman & S.] Range is a field F and P is F-linear. Range is a field F and P is F-linear. Domain is a vector space over F (or some field K extending F). Domain is a vector space over F (or some field K extending F). Property is invariant under affine (sometimes only linear) transformations of domain. Property is invariant under affine (sometimes only linear) transformations of domain. “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” December 2, 2009 IPAM: Invariance in Property Testing 21

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Example: Degree d polynomials Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). #dimensions · d/(K - 1) #dimensions · d/(K - 1) Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. December 2, 2009 IPAM: Invariance in Property Testing 22

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Some results If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests (in weak form) with single proof. Unifies previous algebraic tests (in weak form) with single proof. December 2, 2009 IPAM: Invariance in Property Testing 23

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Some results If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests with single proof. Unifies previous algebraic tests with single proof. If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) December 2, 2009 IPAM: Invariance in Property Testing 29

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Some results If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Linear invariance when P is not F-linear: Linear invariance when P is not F-linear: Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Nice results due to [Shapira] Nice results due to [Shapira] December 2, 2009 IPAM: Invariance in Property Testing 30

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Conclusions Invariance seems to be a very nice perspective on “property testing” … Invariance seems to be a very nice perspective on “property testing” … (Needs Harmonic Analysis ) (Needs Harmonic Analysis ) Hope: Can lead to interesting, new results? Hope: Can lead to interesting, new results? December 2, 2009 IPAM: Invariance in Property Testing 32